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Mirrors > Home > MPE Home > Th. List > elxrge0 | Structured version Visualization version GIF version |
Description: Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
elxrge0 | ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1074 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞)) | |
2 | 0xr 10278 | . . 3 ⊢ 0 ∈ ℝ* | |
3 | pnfxr 10284 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | elicc1 12412 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))) | |
5 | 2, 3, 4 | mp2an 710 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)) |
6 | pnfge 12157 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
7 | 6 | adantr 472 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞) |
8 | 7 | pm4.71i 667 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ↔ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞)) |
9 | 1, 5, 8 | 3bitr4i 292 | 1 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 class class class wbr 4804 (class class class)co 6813 0cc0 10128 +∞cpnf 10263 ℝ*cxr 10265 ≤ cle 10267 [,]cicc 12371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-i2m1 10196 ax-1ne0 10197 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-icc 12375 |
This theorem is referenced by: 0e0iccpnf 12476 ge0xaddcl 12479 ge0xmulcl 12480 xnn0xrge0 12518 xrge0subm 19989 psmetxrge0 22319 isxmet2d 22333 prdsdsf 22373 prdsxmetlem 22374 comet 22519 stdbdxmet 22521 xrge0gsumle 22837 xrge0tsms 22838 metdsf 22852 metds0 22854 metdstri 22855 metdsre 22857 metdseq0 22858 metdscnlem 22859 metnrmlem1a 22862 metnrmlem1 22863 xrhmeo 22946 lebnumlem1 22961 xrge0f 23697 itg2const2 23707 itg2uba 23709 itg2mono 23719 itg2gt0 23726 itg2cnlem2 23728 itg2cn 23729 iblss 23770 itgle 23775 itgeqa 23779 ibladdlem 23785 iblabs 23794 iblabsr 23795 iblmulc2 23796 itgsplit 23801 bddmulibl 23804 xrge0addge 29831 xrge0infss 29834 xrge0addcld 29836 xrge0subcld 29837 xrge00 29995 xrge0tsmsd 30094 esummono 30425 gsumesum 30430 esumsnf 30435 esumrnmpt2 30439 esumpmono 30450 hashf2 30455 measge0 30579 measle0 30580 measssd 30587 measunl 30588 omssubaddlem 30670 omssubadd 30671 carsgsigalem 30686 pmeasmono 30695 sibfinima 30710 prob01 30784 dstrvprob 30842 itg2addnclem 33774 ibladdnclem 33779 iblabsnc 33787 iblmulc2nc 33788 bddiblnc 33793 ftc1anclem4 33801 ftc1anclem5 33802 ftc1anclem6 33803 ftc1anclem7 33804 ftc1anclem8 33805 ftc1anc 33806 xrge0ge0 40061 |
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